Average Error: 3.5 → 1.2
Time: 7.6m
Precision: 64
Internal Precision: 128
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 1.3448150596446022 \cdot 10^{+156}:\\ \;\;\;\;\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2}{\frac{\left(\left(\alpha + \beta\right) + \alpha \cdot \beta\right) + 1.0}{\left(\alpha + \beta\right) + 2}}}}{2 + \left(1.0 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(1.0 + \left(\alpha + \beta\right)\right)} \cdot \frac{1}{\frac{\beta}{\alpha} + \left(\frac{\alpha}{\beta} + 2\right)}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if beta < 1.3448150596446022e+156

    1. Initial program 1.3

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    2. Simplified1.2

      \[\leadsto \color{blue}{\frac{\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity1.2

      \[\leadsto \frac{\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{\color{blue}{1 \cdot \left(2 + \left(\beta + \alpha\right)\right)}}}{2 + \left(\beta + \alpha\right)}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\]

    5. Applied *-un-lft-identity1.2

      \[\leadsto \frac{\frac{\frac{\color{blue}{1 \cdot \left(1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)\right)}}{1 \cdot \left(2 + \left(\beta + \alpha\right)\right)}}{2 + \left(\beta + \alpha\right)}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\]

    6. Applied times-frac1.2

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{1} \cdot \frac{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}}{2 + \left(\beta + \alpha\right)}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\]

    7. Applied associate-/l*1.3

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{1}}{\frac{2 + \left(\beta + \alpha\right)}{\frac{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}}}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\]

    8. Simplified1.3

      \[\leadsto \frac{\frac{\color{blue}{1}}{\frac{2 + \left(\beta + \alpha\right)}{\frac{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\]

    if 1.3448150596446022e+156 < beta

    1. Initial program 15.7

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    2. Simplified15.7

      \[\leadsto \color{blue}{\frac{\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity15.7

      \[\leadsto \frac{\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{\color{blue}{1 \cdot \left(2 + \left(\beta + \alpha\right)\right)}}}{2 + \left(\beta + \alpha\right)}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\]

    5. Applied *-un-lft-identity15.7

      \[\leadsto \frac{\frac{\frac{\color{blue}{1 \cdot \left(1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)\right)}}{1 \cdot \left(2 + \left(\beta + \alpha\right)\right)}}{2 + \left(\beta + \alpha\right)}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\]

    6. Applied times-frac15.7

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{1} \cdot \frac{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}}{2 + \left(\beta + \alpha\right)}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\]

    7. Applied associate-/l*15.7

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{1}}{\frac{2 + \left(\beta + \alpha\right)}{\frac{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}}}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\]

    8. Simplified15.7

      \[\leadsto \frac{\frac{\color{blue}{1}}{\frac{2 + \left(\beta + \alpha\right)}{\frac{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}}}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\]
    9. Using strategy rm

    10. Applied div-inv15.7

      \[\leadsto \color{blue}{\frac{1}{\frac{2 + \left(\beta + \alpha\right)}{\frac{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}} \cdot \frac{1}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}}\]

    11. Taylor expanded around -inf 0.6

      \[\leadsto \frac{1}{\color{blue}{\frac{\beta}{\alpha} + \left(2 + \frac{\alpha}{\beta}\right)}} \cdot \frac{1}{\left(1.0 + \left(\beta + \alpha\right)\right) + 2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 1.3448150596446022 \cdot 10^{+156}:\\ \;\;\;\;\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2}{\frac{\left(\left(\alpha + \beta\right) + \alpha \cdot \beta\right) + 1.0}{\left(\alpha + \beta\right) + 2}}}}{2 + \left(1.0 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(1.0 + \left(\alpha + \beta\right)\right)} \cdot \frac{1}{\frac{\beta}{\alpha} + \left(\frac{\alpha}{\beta} + 2\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019050 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0)))